sscm 1703 - persamaan terbitan 1 - tutorial 4

1. DEPARTMENT OF MATHEMATICAL SCIENCES
FACULTY OF SCIENCE
UNIVERSITI TEKNOLOGI MALAYSIA
SSCM 1703 DIFFERENTIAL EQUATIONS TUTORIAL 4
1. Verify that yp is a particular integral of the following differential equations.
(a) y′′ 3y′ + 2y = x + 4; yp = 1
4 (2x + 11):
(b) y′′ + 9y = 3 cos 2x sin 2x; yp = 1
5 (3 cos 2x sin 2x):
2. Find the values of a; b; c if yp is a particular integral of the differential equations.
(a)
d2y
dx2
dy
dx
+ 12y = 2x; yp = ax + b:
(b)
d2y
dx2 + 6
dy
dx
−4x; yp =
+ 8y = xe
(
ax2 + bx
)
e−4x:
3. By using the method of the undetermined coefficients, find the general solution of the following
equations.
(a)
d2y
dx2 + 5
dy
dx
+ 4y = 3 2x. (b)
d2z
dx2 + 4
dz
dx
= 2x.
(c)
d2z
dt2 + 2
dz
dt
+ 3z = t 2t2. (d) 2
d2y
dx2 + 5
dy
dx
3y = 6x2 2x + 1.
5. Solve the following Initial value problems (IVP’s).
(a)
d2x
dt2 + x = 3 2t2; x = 7 and
dx
dt
= 0 when t = 0.
(b)
d2y
dt2 +
dy
dt
+ 2y = 2t + 4; y = 1 and
dy
dt
= 2 when t = 0.
6. Solve the following Initial value problems (IVP’s).
(a) y′′ 2y′ 8y = 3e4t; y(0) = 0; y′(0) = 3.
(b) z′′ z′ 6z = 3e−2x; z(0) = 0; z′(0) = 2:
7. By using the method of the undetermined method, find the general solution of the following
equations.
(a) y′′ y′ 2y = sin 2x. (b) y′′ + 4y = 1
2 cos 4x.
(c) y′′ 4y′ + 3y = 2 cos x + 4 sin x. (d) y′′ 4y′ + 4y = 4 cos 2x.
8. Find the general solution of the following differential equations.
(a) y′′ + y′ 2y = 2x 40 cos 2x. (b) y′′ 4y = 6 + e2x.
(c) y′′ y′ 2y = 6x + 6e−x. (d) y′′ + 4y = sin 2x cos x.
9. Find the general solution of the following differential equations.
(a) y′′ y = (5x + 1)e3x. (b) y′′ 2y′ 2y = x (ex e−x).
(c) y′′ y′ 2y = ex cos x. (d) y′′ + 5y′ + 6y = e−2x sin 2x.
(e) y′′ + 4y′ + 4y = x2e−2x. (f) y′′ + 2y′ + y = x cos x.
10. Find the general solution of the differential equation
d2y
dx2 + 8
dy
dx
+ 16y = x
(
12 e
−4x)
: